Relaxation to equilibrium in models of classical spins with long-range interactions
For a model long-range interacting system of classical Heisenberg spins, we study how fluctuations, such as those arising from having a finite system size or through interaction with the environment, affect the dynamical process of relaxation to Boltzmann-Gibbs equilibrium. Under deterministic spin precessional dynamics, we unveil the full range of quasistationary behavior observed during relaxation to equilibrium, whereby the system is trapped in nonequilibrium states for times that diverge with the system size. The corresponding noisy dynamics, modeling interaction with the environment and constructed in the spirit of the stochastic Landau-Lifshitz-Gilbert equation, however shows a fast relaxation to equilibrium on a size-independent timescale and no signature of quasistationarity, provided the noise is strong enough. Similar fast relaxation is also seen in Glauber Monte Carlo dynamics of the model, thus establishing the ubiquity of what has been reported earlier in particle dynamics (hence distinct from the spin dynamics considered here) of long-range interacting systems, that quasistationarity observed in deterministic dynamics is washed away by fluctuations induced through contact with the environment.
Keywords: Stationary states, Metastable states, Stochastic particle dynamics
1 Introduction and Model of study
Stochasticity is intrinsic to any statistical system constituted by a finite number of interacting degrees of freedom, which is known to induce fluctuations in both static and time-dependent observables of the system, thereby affecting their statistical properties. Statistical fluctuations are indeed inevitable and fundamental, and are present even when the system is isolated from its environment. Different from such fluctuations are the ones induced in the system through its interaction with the external environment. It is evidently of interest to investigate how these two sources of stochasticity interplay in dictating the long-time state of the system, and in particular, in predicting the values of the macroscopic observables the system attains in the stationary state. In this work, we explore the aforementioned issues within the ambit of a model many-particle system comprising classical spins that are interacting with one another through an inter-particle potential that is long-ranged in nature. Namely, the interparticle potential decays rather slowly as a function of the separation between the particles, specifically, as , with being the embedding spatial dimension of the system [1, 2, 3, 4, 5].
Long-range interacting (LRI) systems may be found at all length scales, from atomic to astrophysical, and are known to exhibit a range of physical phenomena that appear counterintuitive when viewed vis-à-vis short-range systems for which the interaction has a finite range. A basic property that distinguishes LRI systems from short-range ones is the violation of additivity, whereby a macroscopic LRI system cannot be divided into independent macroscopic subparts so that thermodynamic quantities referring to the subparts add up to yield the corresponding values for the composite system. While non-additivity results in such unusual features as inequivalent equilibrium ensembles and a non-concave entropy function, more striking is its consequence on dynamical properties, namely, that an isolated LRI system relaxes to equilibrium over a time that diverges with the number of degrees of freedom . Consequently, the larger the system is, the longer is the time that it takes to attain equilibrium, resulting in slowly-evolving nonequilibrium states being directly accessible to experimental observations .
Our model of study consists of globally-coupled classical Heisenberg spins of unit length denoted by , . Expressed in terms of spherical polar angles and , one has . The Hamiltonian of the system is given by
where is an integer. The spin components satisfy
where is the fully antisymmetric Levi-Civita symbol. Here and in the following, we use Roman indices to label the spins and Greek indices to denote the spin components. In Eq. (1), the first term with on the right hand side (rhs) models a ferromagnetic mean-field interaction between the spins. On the other hand, the second term with on the rhs accounts for local anisotropy: for example, (respectively, ) models quadratic (respectively, quartic) anisotropy, and will be referred to below as the quadratic (respectively, the quartic) model. Single-spin Hamiltonian of Heisenberg spins and involving quadratic and quartic terms has been considered previously in the literature, see, e.g., Ref. . The anisotropy term in Eq. (1) lowers energy by having the magnetization vector
pointing in the plane. The length of the magnetization vector is given by . Note that the canonical variables for the -th spin are and . The coupling constant in Eq. (1) has been scaled down by the system size to order to make the energy extensive, in accordance with the Kac prescription . The system (1) is however intrinsically non-additive, since extensivity does not guarantee additivity, although the converse is true. In the following, we set to unity without loss of generality, and also take unity for the Boltzmann constant.
where the dot denotes derivative with respect to time, and the Poisson bracket for the two functions of the spins are given by  , which may be re-expressed as
Using Eq. (4), we obtain the dynamical evolution of the spin components as
Taking the vector dot product of both sides of Eq. (4) with , it is easily seen that the dynamics conserves the length of each spin. Summing Eq. (8) over , we find that is a constant of motion. The total energy of the system is also a constant of motion, and, as such, Eq. (4) models microcanonical dynamics of the system (1). Note that the dynamical setting of Eq. (4) is very different from that involving particles characterized by generalized coordinates and momenta and time evolution governed by a Hamiltonian given by a sum of a kinetic and a potential energy contribution, e.g., that of the celebrated Hamiltonian mean-field (HMF) model . As a result, none of the results, static or dynamic, derived for the latter may be a priori expected to apply to the model (1). From Eqs. (6)-(8), we obtain the time evolution of the variables and as
where , the effective field for the -th spin, is obtained from the Hamiltonian (1) as
Thus, the effective magnetic field has a global and a local contribution, with the former being due to the magnetization set up in the system by the effect of all the spins, and the latter due to the field set up for individual spins by the anisotropy term in the Hamiltonian (1).
The quadratic model was first considered in Ref.  that addressed the equilibrium and relaxational properties of the model. The system was shown to exhibit in Boltzmann-Gibbs (BG) microcanonical equilibrium a magnetized (equilibrium magnetization ) phase at low values of the energy per spin and a nonmagnetized () phase at high values, with a continuous transition between the two occurring at a critical value . It was established that within microcanonical dynamics and for a class of nonmagnetized initial states, there exists a threshold energy , such that in the energy range , relaxation to equilibrium magnetized state occurs over a time that scales superlinearly with [7, 9]. On the other hand, for energies , the dynamics shows a fast relaxation out of the initial nonmagnetized state over a time that scales as logarithm of . The particular initial state that was considered was the so-called waterbag (WB) state, in which the spins have ’s chosen independently and uniformly over the interval and the ’s chosen independently and uniformly over an interval symmetric about , that is, over the interval , with being a real positive quantity. These results, obtained on the basis of numerical integration of the equations of motion, were complimented by an analytical study in the limit of the time evolution, à la a Vlasov-type equation, of the single-spin phase space distribution. The distribution counts the fraction of the total number of spins that have given and values. It was demonstrated that the distribution associated to the WB state is stationary under the Vlasov evolution, but is unstable for energies below and stable for energies above. For finite , the eventual relaxation to equilibrium observed for energies was accounted as due to statistical fluctuations adding non-zero finite- corrections to the Vlasov equation that are at least of order greater than . The WB state that for energies is stationary and stable for an infinite system but which shows a slow evolution for finite exemplify the so-called quasistationary states (QSSs) .
Starting from the aforementioned premises, we pursue in this work a detailed characterization of the relaxational dynamics and its ubiquity in the context of long-range spin models, by considering the model (1) for general values of . We study for general the BG microcanonical equilibrium properties of the model, deriving in particular an expression for the continuous phase transition point , such that the system is in a magnetized phase for lower energies and in a nonmagnetized phase for higher energies. Though not guaranteed for LRI systems , by virtue of the model (1) exhibiting a continuous transition in equilibrium, we conclude by invoking established results  that microcanonical and canonical ensembles are equivalent in equilibrium. Consequently, one may associate to every value of the conserved microcanonical energy density a temperature of the system in canonical equilibrium that guarantees that the average energy in canonical equilibrium equals in the limit . This allows to also derive the phase diagram of the model (1) in canonical equilibrium.
The WB single-spin distribution is non-analytic at , and one may wonder as to whether such a peculiar feature led to quasitationarity in the model reported in Refs. [7, 10]. As a counterpoint, and to demonstrate that quasistationarity is rather generic to the model (1), we consider as initial states suitably smoothened versions of the WB state, the so-called Fermi-Dirac (FD) state, for which the single-spin distribution is a perfectly analytic function, and study its evolution in time. A linear stability analysis of the FD state under the infinite- Vlasov dynamics establishes the existence of a threshold energy value , such that the state is stationary but linearly unstable under the dynamics for energies . For finite , we establish that the relaxation to equilibrium occurs as a two-step process: an initial relaxation from the FD state to a magnetized QSS, followed by a relaxation of the latter over a timescale to BG microcanonical equilibrium. The magnetized QSS has thus a lifetime . For energies , however, the FD state is dynamically stable under the Vlasov evolution, exhibiting for finite a relaxation towards equilibrium over a scale , where the exponent has an essential dependence on . In this case, one concludes observing a nonmagnetized QSS with a lifetime . As for , while one obtains for the value (as opposed to the value for the WB state reported in Ref. ), one observes a linear dependence () for the quartic model. While magnetization turns out to be a useful macroscopic observable to monitor in order to establish the aforementioned relaxation scenario, it does not serve the purpose when considering energies where both the initial FD and the final BG microcanonical equilibrium state are nonmagnetized. Here, by identifying a suitable observable (e.g., and for respectively the quadratic and the quartic model), we show that the relaxation to equilibrium occurs over a timescale that has an dependence distinct from what was observed for magnetization relaxation for energies . Namely, the relaxation time scales as for the quadratic model and as for the quartic model. We may thus conclude for energies the existence of a nonmagnetized QSS with a lifetime that diverges with the system size.
Our next issue of investigation is the robustness of QSSs with respect to fluctuations induced through contact with the external environment. Modelling the environment as a heat bath, previous studies of Hamiltonian particle dynamics (e.g., that of the HMF model) have invoked a scheme of coupling to the environment that allows for energy exchange and consequent noisy Langevin evolution of the system. These studies have suggested a fast relaxation to equilibrium over a size-independent timescale provided the noise is strong enough [12, 13, 14]. In the context of the model (1), in order to assess the effects of noise induced by the external environment, we study a stochastic version of the dynamics (11) that considers the effective field , see Eq. (11), to have an additional stochastic component due to interaction with the environment. The resulting dynamics, built in the spirit of the stochastic Landau-Lifshitz-Gilbert equation well known in studies of dynamical properties of magnetic systems (see Ref.  for a review), reads
where the second term on the right represents dissipation with the real parameter being the dissipation constant, and is a Gaussian white noise with independent components that satisfy
Here, is a real constant that characterizes the strength of the noise. Note that the stochastic dynamics (13) conserves the length of each spin, as does the deterministic dynamics (11). The former models dynamics within a canonical ensemble for which the energy is not conserved during the dynamical evolution, while, as already mentioned earlier, the latter models energy-conserving microcanonical dynamics.
The presence of noise in Eq. (13) makes the state of the system at a given time, characterized by the set of values , to vary from one realization of the dynamics to another, even when starting every time from the same initial condition. Although Eq. (13) has the flavor of Langevin dynamics, it is different in that the noise and dissipation terms are incorporated in a way that it has the desirable feature of keeping the length of each spin to be unity at all times during the dynamical evolution. Since the noise terms in Eq. (13) depend on the state of the system, itself stochastic in nature, the noise is said to be multiplicative in common parlance. As has been argued in Ref. , requiring the dynamics (13) to relax to BG equilibrium at long times fixes the constant to be related to in the manner
a choice we also consider in this work. Our numerical simulation of the dynamics (13) follows the scheme detailed in Appendix B. The results show that in presence of strong-enough noise, the system shows a fast relaxation to BG equilibrium on a size-independent time scale, with no existence of intermediate quasistationary states.
An alternative way of modeling the effect of environment-induced noise in the dynamics is to invoke a Monte Carlo update scheme of the spin values that guarantees that the long-time state of the system is BG equilibrium. Our investigation of noise effects through an analysis of the dynamics (13) is complemented by a study based on Glauber Monte Carlo dynamics of the system (1). In this scheme, randomly selected spins attempt to rotate by a stipulated amount (which itself could be random) with a probability that depends on the change in the energy of the system as a result of the attempted update of the state of the system [17, 18]. Specifically, to perform the Monte Carlo dynamics at temperature , one implements the following steps :
One starts with a spin configuration in the nonmagnetized FD state.
Next, one selects a spin at random and attempts to change its direction at random, that is, choose a value of uniformly in and a value of uniformly in and assign these values to the spin.
One then computes , the change in the energy of the system that this attempted change of spin direction results in.
If , that is, the system energy is lowered by the change of spin direction, the change is accepted.
On the other hand, if the energy increases by changing the spin direction, that is, , one computes the Boltzmann probability . Next, if a random number chosen uniformly in satisfies , the change in spin direction is accepted; otherwise, the attempted change is rejected and the previous spin configuration is retained.
Time is measured in units of Monte Carlo steps (MCS), where one step corresponds to attempted changes in spin direction.
At the end of every MCS, one computes the desired physical quantities such as the magnetization. In practice, one repeats steps (ii) – (v) to obtain values as a function of time of these physical quantities averaged over a sufficient number of independent configurations.
On implementing the above scheme, we find similar to the study of the dynamics (13) a fast relaxation to BG equilibrium on a size-independent timescale. Our studies thus serve to reaffirm what has been observed earlier in particle dynamics of LRI systems, namely, that quasistationarity, observed in conservative dynamics, is completely washed away in presence of stochasticity in the dynamics.
The paper is organized as follows. In Section 2, we derive the equilibrium properties of the model (1). This is followed in Section 3 by a study of the Vlasov equation and in particular, linear stability of the FD state that represents a stationary solution of the Vlasov equation. Here, we demarcate for two representative values of (namely, ) regions in the parameter space where the state is stable under the Vlasov evolution, thus suggesting the existence of QSSs during relaxation to equilibrium for finite . In Section 4, we report for results from numerical integration of the noiseless dynamics (11), demonstrating in particular the existence of QSSs and their associated timescales for relaxation to equilibrium. In this section, we also present results on the noisy dynamics (13) as well as those obtained from the Glauber Monte Carlo dynamics of the model (1). We draw our conclusions in Section 5. Some of the technical details in the computation of the Vlasov-stability of the FD state are relegated to the Appendix A, while Appendix B details the numerical scheme employed in integrating the stochastic dynamics (13).
2 Equilibrium properties
In this section, we investigate the properties of the system (1) in the thermodynamic limit and in canonical equilibrium at temperature . Note that model (1) is a mean-field system that describes the motion of a spin moving in a self-consistent mean-field generated by its interaction with all the spins, with the single-spin Hamiltonian given by
Consequently, it is rather straightforward to write down exact expressions for the average magnetization and the average energy in equilibrium and in the thermodynamic limit. With , so that the system orders in the -plane, we may choose the ordering direction to be along without loss of generality, yielding . We thus obtain the average equilibrium magnetization along , denoted by , as 
while the average energy per spin equals
From the fact that the model (1) with shows a continuous phase transition in magnetization across critical inverse temperature , we may anticipate that so is the case for general . Consequently, we may consider Eq. (17) close to the critical point, i.e., for , when is small so that the equation may be expanded to leading order in , as
With , one obtains as the value of that sets the bracketed quantity to zero; on performing the integrals, one obtains to be satisfying
Here, is the Gamma function, while is the upper incomplete Gamma function. At the critical point, when , one obtains the critical energy density as , that is,
on performing the integrals, we get
Note that for , one may check using the above expressions that and that , where is the error function, as was reported in Ref. .
Since the phase transition exhibited by the model (1) is a continuous one, the canonical and microcanonical ensemble properties in equilibrium would be equivalent , and hence, Eq. (23) also gives the conserved microcanonical energy density at the transition point. Figure 1 shows for the energy density as a function of , obtained by first solving numerically for a given the transcendental equation (21) for and then using the obtained value of in Eq. (23). Moreover, one may construct a one-to-one mapping between a value of microcanonical equilibrium energy density and canonical equilibrium temperature by first solving Eq. (17) at a given to obtain the equilibrium magnetization , then substituting in Eq. (18) to obtain the corresponding average energy in canonical equilibrium, and finally demanding that the latter is the conserved energy density in microcanonical equilibrium. On carrying out this program for and , one obtains the results shown in Fig. 2, where we also show as a function of microcanonical energy density .
3 Relaxation to equilibrium and quasistationarity
We now discuss how the model (1) in the limit and under the dynamical evolution (11) relaxes to equilibrium while starting far from it, by invoking the corresponding Vlasov equation. The relaxation may be characterized by monitoring the time evolution of the single-spin distribution function , defined such that gives the probability to find a spin at time that has its -component between and and its azimuthal angle between and . With and being canonical coordinates, the flow in the phase space is divergence free, so that conservation of probability implies vanishing of the total time derivative of : . As explained in Ref. , one may more conveniently use a function related to the function as . This new function is defined such that is the probability to have a spin at time with its angles between and and between and . Using the time evolution equation for and Eqs. (9) and (10), one obtains the so-called Vlasov equation for time evolution of as
Let us consider as a far-from-equilibrium initial condition a nonmagnetized state , i.e., with . Such a state may be obtained by sampling independently for each of the spins the angle uniformly in and the angle according to a given distribution that is symmetric about over an interval . We now analyze the dynamical stability of this state with respect of fluctuations, as may be induced by having a finite value of . The method of analysis follows closely the one pursued in Ref. . To this end, we linearize the Vlasov equation (24) with respect to fluctuations by expanding as
Here, since the initial angles of the spins are sampled independently, the small parameter is of order , while we have . The linearized Vlasov equation reads
where and are linear in : .
Now, since is -periodic in , we may implement the following Fourier expansion:
In the long-time limit, we may expect the linearized Vlasov dynamics to be dominated by the Fourier mode of frequency with the smallest imaginary part, so that one effectively has , yielding
with . It then follows that the relevant eigenmodes of Eq. (27) are those with .
Multiplying both sides by and then integrating over , we find, by using the definition of the quantity and the fact that , that
Let us consider as a representative example for the form
where with and being real parameters, and is the normalization constant. In the limit , it is easy to see that is a uniform distribution over the range ; correspondingly, the distribution (32) becomes
and is thus identical to the WB state . For finite but large , the distribution is smoothened around the boundaries at . Figure 3 shows for different values of and for , which makes it evident the similarity in the form of to the Fermi-Dirac (FD) distribution. Henceforth, we will refer to the distribution (32) as the FD state. While it is not possible to derive analytical results for the FD distribution for general , simplifications occur for large when exact expressions may be derived, as detailed below.
In Appendix A, we show that for large , we have
correct to order , while to same order, the energy corresponding to the state (32) is given by
where is defined in Appendix A and is to be obtained by solving Eq. (35). The latter equation gives for two possible values of given by
and for a single value given by
where we have and . On physical grounds, we would want Eqs. (39), (40), and (41) to be valid for all , including . Equation (39) however gives for an inconsistent relation , and hence, may be discarded.
It then follows that the WB state is linearly unstable under the Vlasov dynamics for energy density smaller than , and is linearly stable for . For , the perturbation grows exponentially in time. Here, on setting with real , one gets , so that the magnetization behaves as
Thus, for , the relaxation time over which the magnetization acquires a value of scales as . For , relaxation of the Vlasov-stable WB state towards the BG equilibrium in a finite system occurs as resulting from finite- nonlinear corrections to the Vlasov equation . In this case, the WB state manifests itself in a finite system as a long-lived QSS that relaxes to BG equilibrium over a timescale that diverges with .
For finite but large , when Eqs. (40) and (41) are valid, we may expect on the basis of the above that there exists an energy threshold such that the FD state is linearly unstable and that the scaling (46) holds for energy , while the state manifests itself as a QSS for energies . Such an may be obtained by setting in Eqs. (40) and (41); there are more than one value of for given and that one obtains in doing so, and we take for the physically meaningful only the value that reduces to Eqs. (44) and (45) as one takes the limit . The result for as a function of is shown in Fig. 1 for .
4 Numerical results
Here, we discuss numerical results in support of our theoretical analysis of the preceding section. We start with resenting results for the deterministic dynamics (11), for two representative values of , namely, . In performing numerical integration of the deterministic dynamics, unless stated otherwise, we employ a fourth-order Runge-Kutta integration algorithm with timestep equal to . In the numerical results that we present, data averaging has been typically over several hundreds to thousand runs of the dynamics starting from different realizations of the FD state (32).
We first discuss the results for , for which we make the choice that yields the equilibrium critical energy . Choosing as an initial condition the nonmagnetized FD state (32) with , for which one has the stability threshold (see Fig. 1), Fig. 4(a) shows for energy a fast relaxation out of the initial state on a timescale (see Fig. 4(b)). This is consistent with the prediction based on Eq. (46), which is further validated by the collapse of the data for vs. for different values of shown in Fig. 4(c); here, the growth rate of is obtained as the magnitude of imaginary part of the root of Eq. (40) for which the imaginary part is the largest in magnitude. Figure 5(a) shows that the relaxation observed in Fig. 4 out of the initial FD state is not to BG equilibrium but is to a magnetized QSS that has a lifetime that scales linearly with , Fig. 5(b). Summarizing, for energy , relaxation of nonmagnetized FD state to BG equilibrium is a two-step process: in the first step, the system relaxes over a timescale to a magnetized QSS, while in the second step, this QSS relaxes over a timescale to BG equilibrium.
For energies , Fig. 6(a) shows that consistent with the analysis of Section 3, the initial FD state appears as a nonmagnetized QSS that relaxes to BG equilibrium over a time which by virtue of the data presented in Fig. 6(b) may be concluded to be scaling with as . For energies too is the initial FD state a stable stationary solution of the Vlasov equation, and is expected to show up as a QSS. However, here the magnetization is not the right quantity to monitor since both the FD state and BG equilibrium are nonmagnetized. Consequently, we choose to monitor as a function of time (note that for , the quantity is strictly a constant for infinite , showing fluctuations about this constant value for finite ). Figure 7 shows that indeed the initial FD state does show up as a QSS that has a lifetime that scales quadratically with .
Having discussed the deterministic dynamics, we now turn to results obtained from numerical integration of the stochastic dynamics (13). For the results reported in this work, we take and integration timestep equal to . Data averaging has been typically over several hundreds to thousand runs of the dynamics starting from different realizations of the FD state (32). Our aim here is to compare stochastic dynamics results with those from deterministic dynamics observed at a given energy . By virtue of equivalence of microcanonical and canonical ensembles in equilibrium, we choose the temperature in the stochastic dynamics to have a value that ensures that one has in equilibrium the same value of magnetization as the one observed for the deterministic dynamics at energy ; this is done by using plots such as those in Fig. 2. Figure 5(c) shows that under stochastic dynamics, the initial FD state shows a fast relaxation to BG equilibrium on a size-independent timescale and there is no sign of quasistationarity during the process of relaxation. Figure 5(d) shows that at the chosen value of , the average energy of the system in equilibrium does coincide with the conserved energy of the deterministic dynamics, as it should due to our choice of . Relaxation on a size-independent timescale is also observed for energies (see Fig. 6(a)) and for energies (see Fig. 7(a)). As a further confirmation of the nonexistence of QSS during relaxation to equilibrium, we show in Fig. 8 the results from Glauber Monte Carlo simulation of the system (1).
Referring to Figs. 5,6,7, the relevant timescales to compare between the deterministic and the stochastic dynamics are and , the lifetime of the QSS observed in deterministic dynamics. Indeed, if , the system shows a fast relaxation to equilibrium on a size-independent timescale, while QSSs are observed for satisfying . In particular, in the limit of very large , even a small amount of noise (a very small ) suffices to wash away signatures of quasistationarity.
To demonstrate that the aforementioned relaxation scenario is quite generic to the model (1), we now present in Figs. 9 – 13 results for another value of , namely, . As may be observed from the figures, one has the same qualitative features of the relaxation process as that discussed above for . Note that for energies , one has in contrast to the case the quantity a constant in time and consequently one monitors as a function of time, see Fig. 12. Differences from the case appear in specific scalings of QSSs: the nonmagnetized QSS occurring for energies has a lifetime scaling as , while the one occurring for energies has a lifetime growing with as .
In this work, we wanted to assess the effects of stochasticity, such as those arising from the finiteness of system size or those due to interaction with the external environment, on the relaxation properties of a model long-range interacting system of classical Heisenberg spins. Under deterministic spin precessional dynamics, we showed for a wide range of energy values a slow relaxation to Boltzmann-Gibbs equilibrium over a timescale that diverges with the system size. The corresponding noisy dynamics, modeling interaction with the environment and constructed in the spirit of (i) the stochastic Landau-Lifshitz-Gilbert equation, and (ii) the Glauber Monte Carlo dynamics, however shows a fast relaxation to equilibrium on a size-independent timescale, with no signature of quasistationarity. Our work establishes unequivocally how quasistationarity observed in deterministic dynamics of long-range systems is washed away by fluctuations induced through contact with the environment.
In the light of results on slow relaxation to equilibrium reported in this work, it would be interesting to address the issue of how the system (1) prepared either in BG equilibrium or in QSSs responds to an external field. One issue of particular relevance is when the field is small, and one has for short-range systems in equilibrium a linear response to the field that may be expressed in terms of fluctuation properties of the system in equilibrium. While investigation of similar fluctuation-response relations for LRI systems has been pursued in the context of particle dynamics (e.g., that of the HMF model [20, 21]) and strange scaling of fluctuations in finite-size systems has been reported , it would be interesting to pursue such a study for the spin model (1). Investigations in this direction are under way and will be reported elsewhere .
The normalization satisfies
with , and where in obtaining the last equality, we have performed integration by parts. When is large, the first term on the rhs of Eq. (47) drops out. In order to evaluate the second term, using the fact for large , , we expand about , as
Substituting in Eq. (47), we obtain for large that